The linear system 2x₁ − x₂ + x₃ = −1, 2x₁ + 2x₂ + 2x₃ = 4, −x₁ − x₂ + 2x₃ = −5 has the solution (1, 2,−1)^t .

a. Show that ρ(T_j) = √5/2 > 1.

b. Show that the Jacobi method with x⁽⁰⁾ = 0 fails to give a good approximation after 25 iterations.

c. Show that ρ(T_g) = 1/2 .

d. Use the Gauss-Seidel method with x⁽⁰⁾ = 0 to approximate the solution to the linear system to within 10⁻⁵ in the l_∞ norm.